Example R programs and commands
		    18. Partial correlation coefficients

# All lines preceded by the "#" character are my comments.
# All other left-justified lines are my input.
# All other indented lines are the R program output.

# Read the data - TRANSPOSED
# Y     X1      X2      X3      X4
# --------------------------------------------------
data<-scan()
51.4    0.2     17.8    24.6    18.9
72.0    1.9     29.4    20.7    8.0
53.2    0.2     17.0    18.5    22.6
98.3    9.6     35.6    10.6    5.6
74.8    6.3     28.2    8.8     13.1
92.2    10.8    34.7    11.9    5.9
97.9    9.6     35.8    10.8    5.5
88.1    10.5    29.6    11.7    7.8
62.8    0.4     22.3    26.5    14.3
81.6    2.3     37.9    20.0    0.5

dim(data)<-c(5,10)    # coerce data into a matrix -- TRANSPOSED
x <- t(data)          # un-transpose the data to same form as table
varnames <- c("y","x1","x2","x3","x4")  # names of the variables
dimnames(x)<-list(NULL, varnames)     # label the columns (only) by variable
r <- cor(x)           # Simple correlation matrix for all 5 variables

# View the simple correlation matrix, noting that
# the rows and columns inherit variable name labels from cor():
r

	      y         x1         x2         x3         x4
  y   1.0000000  0.8942124  0.9150668 -0.7502887 -0.8451857
  x1  0.8942124  1.0000000  0.6804904 -0.8754808 -0.5783092
  x2  0.9150668  0.6804904  1.0000000 -0.5708813 -0.9678866
  x3 -0.7502887 -0.8754808 -0.5708813  1.0000000  0.3950981
  x4 -0.8451857 -0.5783092 -0.9678866  0.3950981  1.0000000

# PARTIAL CORRELATION COEFFICIENT calculations:
c <- solve(r)         # inverse of the simple correlation matrix r
dprime <- diag(c)     # d' = diagonal part of c -- VECTOR
d1 <- diag(1/sqrt(dprime))   # inverse of square root of d'

# Partial correlation coefficients are the off-diagonal elements of:
p <- - d1 %*% c %*% d1       # note the minus sign
p

	     [,1]       [,2]       [,3]       [,4]       [,5]
  [1,] -1.0000000  0.8519844  0.6334847  0.3750495  0.2973679
  [2,]  0.8519844 -1.0000000 -0.7005274 -0.7172729 -0.4922659
  [3,]  0.6334847 -0.7005274 -1.0000000 -0.7788338 -0.9147358
  [4,]  0.3750495 -0.7172729 -0.7788338 -1.0000000 -0.7980659
  [5,]  0.2973679 -0.4922659 -0.9147358 -0.7980659 -1.0000000

# Use labels to identify the correlation coefficients:
dimnames(p)<-list(varnames,varnames)  # label the rows and columns
p                                     # print the final result

	      y         x1         x2         x3         x4
  y  -1.0000000  0.8519844  0.6334847  0.3750495  0.2973679
  x1  0.8519844 -1.0000000 -0.7005274 -0.7172729 -0.4922659
  x2  0.6334847 -0.7005274 -1.0000000 -0.7788338 -0.9147358
  x3  0.3750495 -0.7172729 -0.7788338 -1.0000000 -0.7980659
  x4  0.2973679 -0.4922659 -0.9147358 -0.7980659 -1.0000000


# REMARK.
#
# The R function -cov2cor() (note the minus sign) performs the
# right operations to  yield partial correlation coefficients,
# if it gets the inverse correlation matrix as input:
#
-cov2cor(solve(cor(x)))    # partial correlation coefficients

	     y         x1         x2         x3         x4
 y  -1.0000000  0.8519844  0.6334847  0.3750495  0.2973679
 x1  0.8519844 -1.0000000 -0.7005274 -0.7172729 -0.4922659
 x2  0.6334847 -0.7005274 -1.0000000 -0.7788338 -0.9147358
 x3  0.3750495 -0.7172729 -0.7788338 -1.0000000 -0.7980659
 x4  0.2973679 -0.4922659 -0.9147358 -0.7980659 -1.0000000